Page 203 - DECO101_MICRO_ECONOMICS_ENGLISH

Micro Economics

Notes Given that X = X + X
1 2
∂ X ∂ X
∂ X = ∂ X =1
1 2
and the MRs of the duopolists need not be the same. Actually if the duopolists are of unequal size
the one with the larger output will have the smaller MR.

Proof: R = pX i
i
P = a+b(X +X ) = f(X ,X )
1
2
1
2
Thus
∂ R ∂ P
i =P+X
∂ X i ∂ X
i i
But,
∂ P = ∂ P = ∂ P = b
∂ X ∂ X ∂ X
1 2
Therefore,

∂ R i = PX ∂ P = P + ()()
b
+
X
∂ X i ∂ X i
i
Given that P > 0 while b < 0, it is clear that the larger X is, the smaller the MR will be. The two
i
duopolists have different costs
C = f (X ) and C = f (X )
2
2
2
1
1
1
The first duopolist maximises his profit by assuming X constant, irrespective of his own decisions,

2
while the second duopolist maximises his profit by assuming that X will remain constant.

1
The first order condition for maximum profits of each duopolist is

∂ R 1 − ∂ C 1 = 0 ⎤
∂ X 1 ∂ X 1 ⎥ ⎥
∂ R 2 − ∂ C 2 = 0 ⎥
∂ X ∂ X ⎥ ⎦
2 2 ...... (1)
Rearranging, we have
∂ R 1 = ∂ C 1 ⎤
∂ X 1 ∂ X 1 ⎥ ⎥
∂ R 2 = ∂ C 2 ⎥
∂ X ∂ X 2 ⎦ ⎥
2 ...... (2)
Solving the first equation of (2) for X we obtain X as a function of X , that is, we obtain the

1
2
1

reaction curve of firm A. It expresses the output which A must produce in order to maximise his
profit for any given amount X of his rival.

2
Solving the second equation of (2) for X we obtain X as a function of X , that is, we obtain the
2 2 1
reaction function of fi rm B.
If we solve the two equations simultaneously, we obtain the Cournot equilibrium, the values of X
1
and X which satisfy both equations; this is the point of intersection of the two reaction curves.
2
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